On the site percolation threshold of circle packings and planar graphs

TL;DR

This paper establishes the existence of a non-trivial site percolation threshold for circle packings and planar graphs, showing that below a certain probability, infinite clusters do not form, with implications for planar graph percolation.

Contribution

It proves the existence of a positive percolation threshold for circle packings and applies these results to planar graphs, extending percolation theory in geometric and graph-theoretic contexts.

Findings

Percolation with probability p has no infinite cluster on recurrent planar triangulations.

Percolation with probability 1-p has an infinite cluster on transient planar triangulations.

Results support conjectures relating percolation thresholds to graph recurrence and transience.

Abstract

A circle packing is a collection of disks with disjoint interiors in the plane. It naturally defines a graph by tangency. It is shown that there exists $p>0$ such that the following holds for every circle packing: If each disk is retained with probability $p$ independently, then the probability that there is a path of retained disks connecting the origin to infinity is zero. The following conclusions are derived using results on circle packings of planar graphs: (i) Site percolation with parameter $p$ has no infinite connected component on recurrent simple plane triangulations, or on Benjamini--Schramm limits of finite simple planar graphs. (ii) Site percolation with parameter $1-p$ has an infinite connected component on transient simple plane triangulations with bounded degree. These results lend support to recent conjectures of Benjamini. Extensions to graphs formed from the packing of shapes other than disks, in the plane and in higher dimensions, are presented. Several conjectures and open questions are discussed.